House Democrats want to include language that ensures health standards for facilities holding migrants at the border, limits the number of days children can spend in influx facilities and demands that any deaths of migrant children be reported within 24 hours.

But McConnell, speaking from the Senate floor, called the initial House bill "way to the left of the mainstream" and argued that further changes being asked for by House Democrats would "dig back into that abolish ICE [Immigration and Customs Enforcement] playbook and throw a far-left partisan wrench into the whole thing"

"Let me be perfectly clear, I'm glad the Speaker and the administration are discussing some of these outstanding issues. But if House Democrats send the Senate back some partisan effort to disrupt our bipartisan progress, we will simply move to table it," he said.

McConnell added to reporters after his floor speech that the Senate would strip the House changes out of the bill if Democrats send them a revised bill. They would then send the original Senate bill back to the House, where leadership has argued they won't move it to the floor.

The back-and-forth comes as lawmakers had hoped to get a bill for new border funding to Trump's desk by the July Fourth recess, leaving them roughly a day to craft an agreement.

The White House has threatened to veto the House bill, which Senate Republicans believe gives them leverage in trying to force Pelosi to take up their bill.

Though the two bills aligned closely on the top-line figure of more than $4.5 billion, they had deep divisions over issues like Defense Department funding, ICE and reinstating aid for Central American countries.